l05 binary choice models cie555 0205&10
TRANSCRIPT
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CIE 555: Discrete Choice Analysis
Instructor: Qian Wang
02/05&02/10, 2015CIE 555: Discrete Choice Analysis L051
Lecture 5 Binary Choice Models
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Outline
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Binary Choices
Binary Choice Models
Probability of A Choice
Estimation of Binary Choice Models
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An Example of Binary Choices
EZ-Pass payment methods
Choice problem: choose toll payment methods
given the time of day pricing
Decision makers: travelers
Choice alternatives
EZ-Pass vs. Cash
Choice rule: utility
maximization
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Binary Choices
Choice set: C n={alter. 1, alter. 2}
Choice tree
Examples Travel decisions: travel or not
Payment types: paying tolls by E-ZPass or cash
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Alter. 1 Alter. 2
Travel decision
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Binary Choice Models Utility function
Systematic component
Where: = the coefficient of independent variable k ;
= independent variable k for alternative i perceived by
decision maker n.5
nnn V U 111
Systematic utility
Error
nnn V U 222
Alter. 1:
Alter. 2:
Kn K nnn x x xV 112211101 ... Alter. 1:
Kn K nnn x x xV 22222112 '...'' Alter. 2:
', k k
ikn x
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Notations of the Parameters Conceptually: the parameters should vary by
alternatives and decision makers
Ways to deal with the variations of parameters:
Regarding the variations by alternatives:
Try the same parameters at first; if the variable prove tobe significant, vary the parameters by alternatives
Regarding the variations by decision makers:
Split the whole population to several homogeneous
groups, and then specify the choice models for each
group; OR Treat the parameters as random variables that vary
across individuals (random parameter utility functions)
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Error Terms
We assume the distributions of the error terms inorder to calculate the probability of a choice
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Probability of A Binary Choice
The probability of choosing alternative i (i =1 or 2)
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}]2,1{,Pr[
}]2,1{,Pr[)|(
jV V
jV V C i P
jninin jn
jn jnininn
Let: in jnn
We get:
}]2,1{,Pr[)|( jV V C i P jninnn
Its distribution is the key to
calculate the probability
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Linear Probability Model
Assuming the difference of errors follows anuniform distribution
9
-L L
Density function of the error
difference
in jnn
L2
1
:)( n f
L L L
Lor L
f n
nn
n
,2
1
,0
)(
0
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Linear Probability Model (Cont.)
The probability of choosing alter. i :
10
jnin V V
L
jnin
jnin
jnin
jnin
nnn
LV V
LV V L L
LV V
LV V
d f i P
,1
,2
,0
)()(
-L L
Probability of choosing alter. i
jnin V V
:)(i pn
0.5
1
0
045
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Linear Probability Model (Cont.)
The value of L:
The probability only depends on the ratio of the
utility difference to L but not L
We can arbitrarily set L=1/211
LV V L L
LV V i p jnin
jnin
n
,2
)(
Let: 11, LV V jnin
We get:
11,
2
1
2
)(
L
L Li pn
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Binary Probit Model
In reality, few cases match the uniformdistributions
Since the error terms represent the combination
of various sources of randomness, by the central
limit theorem, their distributions would tend to be
normal
Note: central limit theorem states that the re-
averaged sum of a sufficiently large number of
identically distributed independent random
variables, each with finite mean and variance, will
be approximately normally distributed
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Binary Probit Model (Cont.)
Assuming the difference of errors follows annormal distribution with mean as zero and
variance as
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]2
1exp[
2
1)(
2
nn f
)( n f (Density function)
0n
2
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Binary Probit Model
The probability of choosing alter. i :
14
)(
]2
1exp[
2
1
]2
1exp[
2
1
)()(
/)(2
2
jnin
V V
V V
n
V V
nnn
V V
d
d
d f i P
jnin
jnin
jnin
Standardized cumulative normal
distribution function
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Binary Probit Model
The probability curve
The value of :
We choose15
Probability of choosing alter. i
jnin V V
:)(i pn
0.5
1
1
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Binary Logit Model
Assuming the difference of errors follows anlogistic distribution
It is “probitlike”: it resembles the normal
distribution in shape but has heavier tails
It is analytically convenient
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Binary Logit Model (Cont.)
The probability density function
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nnn
n
e
e f
,0,)1(
)(2
Where:= a positive scale
parameter
s = set as 1 for the binary
logit case
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Binary Logit Model (Cont.)
The probability of choosing alter. i :
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jnin
in
jnin
jnin
n
n
jnin
V V
V
V V
V V n
V V
nnn
ee
e
e
d ee
d f i P
)(
2
1
1
)1(
)()(
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Binary Logit Model (Cont.)
The probability curve
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Probability of choosing alter. i
jnin V V
:)(i pn
0.5
1
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Binary Logit Model (Cont.)
Replace V by the utility functions
This indicates that the scale parameter ( )
cannot be distinguished from the overall scales of
parameters ( ) For convenience, we assume that
Furthermore, this implies that
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)(
)(
1
1
1
1)(
jnin
jnin
X X
V V n
e
ei P
1
3/)var( 2 n
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Extreme Cases of the Linear, Probit
and Logit Models
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Probability of choosing alter. i
jnin V V
:)(i pn
0.5
1 0,0, L
L,,0
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Other Binary Choice Models
Arctan probability model
Right-truncated exponential model
Left-truncated exponential model
Reference: Section 4.2 of the textbook (page 73)
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Estimation of Binary Choice Models Model estimation procedure
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Specifying utility functional
forms
Estimating parameters
Validation
Application (Forecasting)
Calibration:
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Methods to Estimate Parameters
Maximum likelihood estimation Likelihood: the probability that the whole
population/sample of decision makers make certain
choices
Methodology: find the best parameter values thatmaximize the logarithm likelihood
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Maximum Likelihood Estimation Likelihood function:
The log likelihood becomes
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)(
1
2
1,21
)ln(
)ln()...,,(
i N n
ni
N
n i nini K
p
p y LL
N
n i
y
ni K ni p L
1
2
1
,21 )...,,(
y ni =1 if decision maker n
chooses alternative i ; 0,
otherwise
Probability for decisionmaker n to choose
alternative i
The group of decision makers
who chose alternative i
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Maximum Likelihood Estimation
Step 2: Find the optimal solutions of theparameters by maximizing the likelihood function
Step 2.1: Find the POTENTIAL optimal solutions by
solving the first-order conditions (first-order
derivatives):
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)...,,(max ,21 K LL
Given unknown variables: K ,...,, 10
K k i p
i p y
LL N
n i n
k nin
k
,...,0,0)(
/)(
1
2
1
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Maximum Likelihood Estimation Step 2.2: Pick the optimal solution by checking the
Hessian matrix of the likelihood function (second-
order condition)
If it is a negative semidefinite matrix: the likelihood function
is concave with a UNIQUE optimal solution
If not: calculate the likelihood value for each solution from
the first-order condition, and choose the solution resulting in
the maximum value of likelihood
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][
2
l k
LL Hessian
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Example: Estimation of Binary
Logit Models
In the binary Logit Models, the probabilities are:
Step 1: the log likelihood function
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jnin
jn
jnin
in
X X
X
n X X
X
nee
e j p
ee
ei p
''
'
''
'
)(,)(
N
n X X
X
in X X
X
in
K
jnin
jn
jnin
in
eee y
eee y
LL
1''
'
''
'
,21
)]log()1()log([
)...,,(
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Example: Estimation of Binary
Logit Models (Cont.)
Step 2: Maximum likelihood problem
Step 2.1: Find the POTENTIAL optimal solutions by
solving the first-order conditions (first-order
derivatives):
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)...,,(max ,21 K LL
Given unknown variables: K ,...,, 10
K k xi p y LL
N
n
nk nin
k
,...,1,0)]([1
f
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Example: Estimation of Binary
Logit Models (Cont.) Step 2.2: Pick the optimal solution by checking the
Hessian matrix of the likelihood function (second-
order derivatives)
The Hessian matrix is negative semidefinite (check page 85
in the textbook to see why) the solution from the first-
order condition is the ONLY optimal solution
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N
nnl nk nn
l k x xi pi p
LL
1
2
))(1)((
P i D i d f h Fi
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Properties Derived from the First-
Order Conditions
For the derivatives corresponding to the constantterm:
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0)]([1
0
0
N
n
nnin xi p y LL
=1
}2.,1.{,)(
0)]([
11
1
alter alter ii p y
i p y
N
n
n
N
n
in
N
n
nin
Which is equivalent to:
Property 1
Where: N = the total number of the observations in the full sample
P ti D i d f th Fi t
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Properties Derived from the First-
Order Conditions (Cont.)
For the derivatives corresponding to thealternative-specific dummy variables:
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}''{,0)]([1
K k xi p y LL N
n
nk nin
k
'
1
'
1
'
1
)(
}''{,0)]([
N
n
n
N
n
in
N
n
nin
i p y
K k i p y
Which is equivalent to:
Property 2
0
1
nk x
For a subset of the sample whose x nk equal 1
For the remainder of the sample
Where: N’ = the number of observations in the subset of the sample
C t ti l A t f
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Computational Aspects of
Maximum Likelihood
The parameters are estimated by solving a groupof NOLINEAR equations from the first-order
conditions
Iterative procedures are required to obtain thesolutions, e.g., the Newton-Raphson algorithm
(check the page 82 in the textbook)
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K k xi p y
LL N
nnk nin
k ,...,1,0)]([1
Nonlinear functions of the unknown parameters
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Learning Outcomes
What is a binary (binomial) choice situation What are the three typical choice models to deal
with the situations, and how they are derived
based on what assumptions
Binary-uniform
Binary-logit
Binary-probit
Be able to apply the appropriate method to deal
with a real-world binary choice problem
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References
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Chapter 4 of the text book Holguín-Veras, J., & Wang, Q. (2011). Behavioral
investigation on the factors that determine
adoption of an electronic toll collection system:
Freight carriers. Transportation research part C:Emerging technologies, 19(4), 593-605.
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Next Class
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Lab 1: use LIMDEP to estimate binary choicemodels
Bring your laptop with the installed software to the
class
Lab 1 assignment will be given